Question: Let $f(x)=x^{^{\scriptsize\dfrac{3}{2}}}$. $f'(x)=$
Solution: The derivative of $f$ can be found using the power rule : $\dfrac{d}{dx}(x^n)=n\cdot x^{n-1}$ (Remember that this applies even when $n$ is a fraction.) $\begin{aligned} &\phantom{=}f'(x) \\\\ &=\dfrac{d}{dx}\left(x^{^{\frac{3}{2}}}\right) \\\\ &=\dfrac{3}{2}x^{^{\frac{3}{2}-1}} \gray{\text{The power rule}} \\\\ &=\dfrac{3}{2}x^{^{\frac{1}{2}}} \end{aligned}$ In conclusion, we found that $f'(x)=\dfrac{3}{2}x^{^{\frac{1}{2}}}$. This can also be written as $1.5\sqrt x$ (all equivalent forms are accepted).